Download work - energy - Gonzaga Physics Department

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WORK - ENERGY
Object
To investigate the work-kinetic energy relationship and conservation of energy.
Apparatus
Track and associated stops, one dynamics cart, PVC tube, one photogate timer, one force sensor, interface
equipment, masses, calipers, meterstick and a mass balance.
Theory
The work kinetic energy relationship gives us a new way, other thanforces and kinematics, to understand physical
systems. One does work by exerting a force, on a moving object, along the line of that object’s motion. Work
is an attempt to change the speed of an object. If a force is exerted at some angle with respect to the direction
of motion, only the component of that force along the line of motion does work. If a constant force F acts on an
object as it undergoes some displacement s along a straight line such that θ is the angle between the direction
of F and the direction of s then the work done by that force on the object is given by
Work = F s cos (θ) = F|| s
(1.1)
If the parallel component of the force F is not remaining constant, then calculus, or graphical techniques, must
be used to determine the work done by a force. If the work done by each individual force acting on an object
as it undergoes some displacement, can be found, these individual works may be added together, as scalers,
to determine the net work done on an object. Rather, one could first add the forces together, as vectors, to
determine the net force acting on an object during some displacement and the work done by this net force would
also equal the net work done on an object.
W orknet
=
X
Worki
= Fnet s cos (θ)
(1.2)
i
The net work done on an object, during any motion, is equal to the change in the object’s kinetic energy
during that motion. Kinetic energy is a measure of an object’s ability to do work, stored in the motion of that
object. Kinetic energy depends on the mass of the object and the speed with which that object is moving.
W orknet
= KE F inal − KE Initial
=
mvf2
2
−
mvi2
2
(1.3)
This equation is the work-kinetic energy relationship. The work kinetic energy relationship is sometimes written
in a more convenient form for situations involving “conservative” forces. A conservative force is one for which
the work done on an object as it moves from one point to another is independent of the path taken between
those points, rather the work only depends on the initial and final positions. In such a case, the work done by
that conservative force can be expressed in terms of the negative of the change in a potential energy function
associated with that conservative force. Each conservative force has its own potential energy function.
W orkcons = − (P E F inal − P E Initial )
(1.4)
The two most common conservative forces that come up in first semester physics classes are the gravitational
force exerted on a mass by the earth and the force exerted by a spring. The potential energy functions associated
with those forces are
k stretch2
(1.5)
P Egrav = m g y
P E spring =
2
2
where k is the spring constant of the spring. If the net work in equation 1.3 is split into two parts, work done
by conservative forces and work done by nonconservative forces, the work done by conservative forces can all
be accounted for using the changes in the appropriate potential energy functions.
W orknet
= W ork nc + W ork cons = W ork nc − (P E F inal − P E Initial )
(1.6)
The PE terms are understood to refer to the sum of all potential energy functions in the system, one for each
conservative force that does work. Combining this relationship with the work kinetic energy relationship and
regrouping terms one gets
W orknc
= (KE F inal + P E F inal ) − (KE Initial + P E Initial ) = EF inal − EInitial
(1.7)
where we have defined E, the mechanical energy, to be the sum of the potential and kinetic energy of a system.
If there is no work done by nonconservative forces, it follows from equation 1.7 that the initial mechanical energy
must equal the final mechanical energy; i.e. the mechanical energy is conserved if there is no work done by
nonconservative forces.
KEInitial + P E Initial = KE F inal + P E F inal
(1.8)
Let’s see how these relationships can be applied to the physical systems you will be studying during this lab.
Part 1. Cart on an inclined track.
A cart is released from rest on a track inclined at some angle θ with respect to the horizontal. We’ll assume friction and air resistance are negligible. As the cart moves down the incline, only two forces act on it;
gravity and the normal force. Since the normal force acts perpendicular to the motion of the cart, it cannot do
any work on the cart. Only the parallel component of the gravitational force, mgsin(θ), does work on the cart.
If s is the distance the cart moves down the incline then equation 1.3 becomes
W orknet
= W ork grav
= m g sin (θ) s =
mvf2
2
(1.9)
Part 2. Cart on an inclined track pushing pvc tube.
In this case, we consider the same situation as in part 1, however, in addition the cart is pushing a pvc
tube down the incline as well. What forces are doing work as the combination moves down the incline? The
weight force on the cart does work mc gsin(θ) s and the weight force on the pvc tube does work mp gsin(θ) s. A
non-negligible frictional force also acts on the pvc tube. The magnitude of this force is µk N where N is the
normal force on the pvc tube from the incline; N = mgcos(θ). The angle between the direction of this force and
the direction is 180 degrees so the work done by this frictional force is -µk mp gscos(θ) s. The normal forces the
cart and pvc tube exert on each other do work as well, however, the cart does just as much positive work on
the pvc tube as the tube does negative work on the cart. When added together, these two works sum to zero.
Ad a result, the net work done is just the sum of the work done by both gravitational forces and the frictional
force. When plugged into equation 1.3 we get
W orknet
= [(mc + mp ) g sin (θ) − µk mp g cos (θ)] s =
(mc + mp ) vf2
2
(1.10)
Part 3. Cart on an inclined track with string connected to mass.
In this case, the cart will be on the incline connected, via a light string over a pulley at the upper end of
the track, to a mass which may fall towards the ground. As the mass falls, the cart will be pulled up the incline
the same distance s that the cart falls. The pulley may be assumed to be massless and frictionless so that the
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tension everywhere in the string is assumed to be uniform. Any frictional force on the cart from the incline is
assumed to be negligible as is air resistance. In that case, the only non conservative force that may do work
is the tension in the string. However, the tension would do just as much positive work on the cart as it does
negative work on the hanging mass. As a result, the total work done by nonconservative forces will be zero.
The mechanical energy of the system should be conserved as the objects move. If the system is released from
rest, the initial kinetic energy will be zero. As the system begins to move, the mass will move downward and
lose gravitational potential energy. At the same time, the cart will be moving upward thus gaining gravitational
potential energy. If the mass loses more gravitational energy than the cart gains, the difference will be converted
into kinetic energy in the system. Both the mass and the cart will have to move with the same speed. Our
conservation of energy relationship can then be applied to this system.
Procedure
Part 0. Determine the coefficient of friction between the pvc tube and the incline.
1. The track should be leveled on the bench. Measure and record the mass of the pvc tube and your cart. Place
the tube on the track and place the appropriate string around the pvc tube. Securely place an additional 1-2
kg on top of the pvc tube. Record this amount. Connect the force probe to the string so that it may pull the
tube along the track.
2. Setup the DataStudio software so that you can view a digits window and a graph for the force probe output.
Start acquiring data. Zero the force probe and then pull the pvc tube along the level track at as constant a
speed as possible keeping the string parallel to the track. Stop acquiring data.
3. Determine and record the average magnitude of the force probe reading while the tube was being pulled at
a constant speed.
Part 1. Cart on an incline.
1. Set the incline to an angle of inclination of around 8 degrees to the horizontal. Determine θ accurately by
measuring distance and using trig.
2. Set the photogate up near the bottom of the track such that the small tube attached to the cart will block
the beam but nothing will collide with the photogate. Determine the location of the cart when it is centered in
the photogate beam. This is the “final” position of the cart during its motion on the track.
3. Place the cart 1.00 meters from the “final” position on the track. Release the cart from rest and record the
time it blocks the photogate beam. Be sure and catch the cart after it passes the photogate so it does not crash
off of the track.
4. Repeat twice more for a total of 3 measured times for this distance d. Record the distance the cart moved
from release until it reached the final position and the 3 measured times.
5. Repeat at 5 other distances along the track spanning distances from 15 cm up to the maximum possible
distance.
6. Using the drill bits, as in the ballistics lab, collect data and determine the width of your photogate beam.
Use calipers to measure the diameter of the small pvc tube attached to your cart which blocked the photogate
beam. Record this data.
Part 2. Cart and PVC tube on incline.
1. Repeat Part 1 only this time have the large pvc tube on the track in front of the cart being pushed down the
track. Be sure that the photogate is placed such that it will be blocked while the cart passes but not hit by the
large pvc tube in front of the cart. You should acquire 3 measured times at each of 6 different distances just as
in part 2.
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Part 3. Cart on incline with pulley and hanging mass.
1. Attach a pulley to the upper end of the track. Attach a length of string to the cart such that when the cart
is at the lower end of the track the string just extends over the pulley far enough to connect to a mass hanger.
Move the mass hanger to the ground thus pulling the cart up the track. Place the photogate such that it would
have finished being blocked by the cart just before the mass hits the ground. Determine the position of the
cart when it is centered while blocking the photogate beam. This is the “final” location of the cart. Move the
cart back down the incline until the mass hanger is back as close to the pulley as possible. This is the “initial”
location of the cart. Record the distance, s, between these two locations.
2. Attach enough mass to the hanger such that it will cause the cart to move up the track when released from
rest. Place the cart at the initial position and release it from rest. Record the amount of time the cart blocked
the beam. Repeat for two more measurements giving three measured times. Record the total amount of mass
m that was attached to the string over the pulley.
3. Repeat for 5 other attached masses, spanning a range of several hundred grams, always releasing the cart
from rest at the same location. Record three times for each mass.
Calculations
Part 0. Determine the coefficient of friction between the pvc tube and the incline.
1. Use the data from part 0 to calculate the coefficient of kinetic friction between the pvc tube and the incline.
You will use this value as your “accepted” value for the coefficient of kinetic friction between these surfaces in
part 2 calculations.
Part 1. Cart on an incline.
1. Determine the distance that your cart moved while it was blocking the photogate. This distance will be the
same throughout the rest of your calculations.
2. For each of your six distances, calculate the average of your three measured times. Use this time and the
distance the pvc tube moved while blocking the photogate beam to determine the speed of the cart at the “final”
position.
3. Calculate the kinetic energy of the cart at the “final” position. Since the initial kinetic energy of the cart
was 0 Joules, the “final” kinetic energy is the change in kinetic energy.
4. Calculate the work done on the cart by gravity. Do this calculation using your knowledge of the distance the
cart moved, the magnitude of the gravitational force on the cart and the angle of the ramp; not using the work
kinetic energy relationship. Equation 1.1 will be useful here. If we neglect any frictional effects, the work done
by gravity is equal to the net work done on the cart as it moves down the incline.
5. Construct a graph of net work done (work done by gravity) vs the change in kinetic energy. Perform a
linear regression and compare the slope and y-intercept of your best fit line with expected values. You may find
equation 1.9 useful in determining the expected values for the slope and y-intercept.
Part 2. Cart and PVC tube on incline.
1. For each of your six distances, calculate the average of your three measured times. Use this average time to
calculate the speed of the cart and large pvc tube at the “final” position.
2. Calculate the total kinetic energy of the tube and cart at the “final” position.
3. Construct a graph of final kinetic energy vs s. Perform a linear regression and compare the slope and
y-intercept of your best fit line with expected values. You may find equation 1.10 useful in determining the
expected values for the slope and y-intercept. In your lab report, you should show sample calculations for any
expected value that you have to calculate.
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Part 3. Cart on incline with pulley and hanging mass.
1. Determine the average time to block the photogate for each of your six different attached masses. Also
calculate the speed of the cart at the “final” position and the kinetic energy of the system when the cart reaches
the “final” position.
2. Calculate the initial and final potential energy for the cart as it moved up the incline from the “initial” to
the “final” position. This value should be the same for all six attached masses.
3. Calculate the initial and final gravitational potential energy for the hanging mass while the cart moved up
the incline from the “initial” to the “final” position for each of the six different attached masses.
4. Calculate the initial and final mechanical energy for the system. for each of the six setups.
5. Construct a graph of final mechanical energy of the system vs initial mechanical energy of the system. Perform a linear regression and compare the slope and y-intercept with expected values.
Questions
1. In part 3 of the lab, is the mechanical energy of ONLY the cart conserved as it moves along the ramp?
Explain.
2. Is mechanical energy conserved in part 2 of the lab as the objects move down the ramp? If you made a graph
of Ef inal vs Einitial for part 2, would the slope of your best fit line be greater than, less than or equal to 1?
Explain.
3. What could be changed about the procedure or calculations in part 2 such that the expected y-intercept
would not be zero? Explain the physical meaning of your, now, nonzero y-intercept.
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WORK-ENERGY DATA SHEET
Part 0 - Coefficient of friction for PVC Tube
Quantity
PVC Tube Mass
Value
Cart Mass
Added Mass
Average Force
Part 1 - Cart on an incline
Ramp Angle
Trial
Units
1
Distance
Time 1
Time 2
Time 3
1.00
2
3
4
5
6
Part 2 - Cart and PVC tube on an incline
Trial
Units
1
2
3
4
5
6
Distance
Time 1
Time 2
Time 3
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WORK-ENERGY DATA SHEET
Part 3 - Cart on incline with pulley and hanging mass
Distance (s)
Trial
Units
1
2
3
4
5
6
Attached Mass
Time 1
Time 2
Time 3